Problem: $f(x, y) = \left( y, \dfrac{x}{y} \right)$ What is $\dfrac{\partial f}{\partial x}$ ? Choose 1 answer: Choose 1 answer: (Choice A) A $\left(1, \dfrac{-x}{y^2} \right)$ (Choice B) B $\left(0, \dfrac{1}{y} \right)$ (Choice C) C $\left(1, \dfrac{1}{y} - \dfrac{x}{y^2} \right)$ (Choice D) D $\left( 0, \dfrac{-1}{y^2} \right)$
Solution: The partial derivative of a vector valued function is component-wise partial differentiation. $\begin{aligned} &f(x, y) = (f_0(x, y), f_1(x, y)) \\ \\ &f_x = \left( \dfrac{\partial f_0}{\partial x}, \dfrac{\partial f_1}{\partial x} \right) \\ \\ &f_y = \left( \dfrac{\partial f_0}{\partial y}, \dfrac{\partial f_1}{\partial y} \right) \end{aligned}$ Because we're taking a partial derivative with respect to $x$, we'll treat $y$ as if it were a constant. Therefore, $f_x = \left(0, \dfrac{1}{y} \right)$.